the method by the user is needed). There are also several other software packages which may be useful for solving zero-dimensional systems. Some of them are listed after the automatic solvers. The Maple function RootFinding[Isolate] takes as input any polynomial system over the rational numbers (if some coefficients are floating point numbers, they are converted to rational numbers) and outputs the real solutions represented either (optionally) as intervals of rational numbers or as floating point approximations of arbitrary precision. If the system is not zero dimensional, this is signaled as an error. Internally, this solver, designed by F. Rouillier computes first a Gröbner basis and then a Rational Univariate Representation from which the required approximation of the solutions are deduced. It works routinely for systems having up to a few hundred complex solutions. The rational univariate representation may be computed with Maple function Groebner[RationalUnivariateRepresentation]. To extract all the complex solutions from a rational univariate representation, one may use MPSolve, which computes the complex roots of univariate polynomials to any precision. It is recommended to run MPSolve several times, doubling the precision each time, until solutions remain stable, as the substitution of the roots in the equations of the input variables can be highly unstable. The second solver is PHCpack, written under the direction of J. Verschelde. PHCpack implements the homotopy continuation method. This solver computes the isolated complex solutions of polynomial systems having as many equations as variables. The third solver is Bertini, written by D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler. Bertini uses numerical homotopy continuation with adaptive precision. In addition to computing zero-dimensional solution sets, both PHCpack and Bertini are capable of working with positive dimensional solution sets. The fourth solver is the Maple library RegularChains, written by Marc Moreno-Maza and collaborators. It contains various functions for solving polynomial systems by means of regular chains. See also Elimination theory Systems of polynomial inequalities Triangular decomposition Wu's method of characteristic set References Equations Algebra Computer algebra Polynomials Algebraic geometry